A213385 -id:A213385 - cp's OEIS frontend

A002846 Number of ways of transforming a set of n indistinguishable objects into n singletons via a sequence of n-1 refinements.

1, 1, 1, 2, 4, 11, 33, 116, 435, 1832, 8167, 39700, 201785, 1099449, 6237505, 37406458, 232176847, 1513796040, 10162373172, 71158660160, 511957012509, 3819416719742, 29195604706757, 230713267586731, 1861978821637735, 15484368121967620, 131388840051760458

Offset: 1

N. J. A. Sloane. Entry revised by N. J. A. Sloane, Jun 11 2012

Construct the ranked poset L(n) whose nodes are the A000041(n) partitions of n, with all the partitions into the same number of parts having the same rank. A partition into k parts is joined to a partition into k+1 parts if the latter is a refinement of the former.
The partition n^1 is at the left and the partition 1^n at the right. The illustration by Olivier Gérard shows the posets L(2) through L(8).
Then a(n) is the number of paths of length n-1 in L(n) that join n^1 to 1^n.
Stated another way, a(n) is the number of maximal chains in the ranked poset L(n). (This poset is not a lattice for n > 4.) - Comments corrected by Gus Wiseman, May 01 2016

			a(5) = 4 because there are 4 paths from top to bottom in this lattice:
  .
       ooooo
     /      \
  o.oooo   oo.ooo
    |    X    |
  o.o.ooo  o.oo.oo
     \       /
      o.o.o.oo
          |
      o.o.o.o.o
  .
(This is the ranked poset L(5), but drawn vertically rather than horizontally.)

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 1..80
P. Erdős, R. K. Guy and J. W. Moon, On refining partitions, J. London Math. Soc., 9 (1975), 565-570.
R. K. Guy, Letter to N. J. A. Sloane, June 24 1971: front, back [Annotated scanned copy, with permission]
Olivier Gérard, The ranked posets L(2),...,L(8)
Gus Wiseman, Hasse Diagrams of Partition Refinement Posets n=1..9
Gus Wiseman, Hasse Diagrams of Partition Refinement Posets n=1..9, Version 1, [Cached copy, with permission]
Gus Wiseman, Hasse Diagrams of Partition Refinement Posets n=1..9, Version 2, [Cached copy, with permission]

See A213242, A213385, A213427 for related sequences, A327643.

v:= l-> [seq(`if`(i=1 or l[i]>l[i-1], seq(subs(1=[][], sort(subsop(
         i=[j, l[i]-j][], l))), j=1..l[i]/2), [][]), i=1..nops(l))]:
b:= proc(l) option remember; `if`(max(l)<2, 1, add(b(h), h=v(l))) end:
a:= n-> b([n]):
seq(a(n), n=1..30);  # Alois P. Heinz, Sep 22 2019

Mathematica
```
<Mitch Harris, Jan 19 2006 *)
```

def A002846(n): return Posets.IntegerPartitions(n).chain_polynomial().leading_coefficient()  # Max Alekseyev, Dec 23 2015

a(17)-a(25) from Mitch Harris, Jan 19 2006

A265947 Total size of all principal order ideals in the poset of integer partitions of n with the refinement order.

Original entry on oeis.org

1, 1, 3, 6, 14, 26, 55, 99, 192, 340, 619, 1063, 1873, 3129, 5308, 8718, 14385, 23116, 37346, 58949, 93294, 145131, 225623, 345833, 529976, 801675, 1211225, 1811558, 2703327, 3998289, 5901849, 8641160, 12623450, 18315370, 26503133, 38119289, 54691750, 78028166, 111041918, 157250528, 222105633

Offset: 0

Max Alekseyev, Dec 23 2015

nonn

a(n) is the number of refinement-ordered pairs of integer partitions of n. Every such pair (x,y) is a multiset union x and a multiset of sums y of some weakly ordered sequence of integer partitions, so this sequence is dominated by A063834 (twice partitioned numbers). - Gus Wiseman, May 01 2016

			a(4) = 14 ordered pairs of partitions: {(4,4), (4,22), (4,31), (4,211), (4,1111), (22,22), (22,211), (22,1111), (31,31), (31,211), (31,1111), (211,211), (211,1111), (1111,1111)}.

Jon Mark Perry et al., Counting refinements of partitions, Mathoverflow, 2015.

Cf. A001764, A002846, A213242, A213385, A213427, A063834.

def A265947(n):
    P = Posets.IntegerPartitions(n)
    return sum( len(P.order_ideal([p])) for p in P )

# Alternative:
def A265947(n):
    return Posets.IntegerPartitions(n).relations_number() # F. Chapoton, Feb 26 2020

A213427 Number of ways of refining the partition n^1 to get 1^n.

Original entry on oeis.org

1, 1, 2, 6, 18, 74, 314, 1614, 8650, 52794, 337410, 2373822, 17327770, 136539154, 1115206818, 9671306438, 86529147794, 816066328602, 7904640819682, 80089651530566, 832008919174434, 8983256694817802, 99219778649809162, 1134999470682805134, 13241030890523397154

Offset: 1

N. J. A. Sloane, Jun 11 2012

nonn

Consider the ranked poset L(n) of partitions defined in A002846. Add additional edges from each partition to any other partition that is a refinement of it. In L(5), for example, we add edges from 5^1 to 31^2, 2^21, 21^3 and 1^5, from 41 to 21^3 and 1^5, and so on.

Then a(n) is the total number of paths in the augmented poset of any length from n^1 to 1^n.

Alois P. Heinz, Table of n, a(n) for n = 1..40
Olivier Gérard, The ranked posets L(2),...,L(8)

Cf. A002846, A213242, A213385.

b:= proc(l) option remember; local i, j, n, t; n:=nops(l);
      `if`(n<2, {[0]}, `if`(l[-1]=0, b(subsop(n=NULL, l)), {l,
      seq(`if`(l[i]=0, {}[], {seq(b([seq(l[t]-`if`(t=1, l[t],
      `if`(t=i, 1, `if`(t=j and t=i-j, -2, `if`(t=j or t=i-j,
      -1, 0)))), t=1..n)])[], j=1..i/2)}[]), i=2..n)}))
    end:
p:= proc(l) option remember;
      `if`(nops(l)=1, 1, add(p(x), x=b(l) minus {l}))
    end:
a:= n-> p([0$(n-1), 1]):
seq(a(n), n=1..25);  # Alois P. Heinz, Jun 12 2012

More terms from Alois P. Heinz, Jun 11 2012

Edited by Alois P. Heinz at the suggestion of Gus Wiseman, May 02 2016

A381454 Number of multisets that can be obtained by choosing a strict integer partition of each prime index of n and taking the multiset union.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 1, 1, 2, 3, 1, 4, 2, 2, 1, 5, 1, 6, 2, 2, 3, 8, 1, 3, 4, 1, 2, 10, 2, 12, 1, 3, 5, 4, 1, 15, 6, 4, 2, 18, 2, 22, 3, 2, 8, 27, 1, 3, 3, 5, 4, 32, 1, 6, 2, 6, 10, 38, 2, 46, 12, 2, 1, 8, 3, 54, 5, 8, 4, 64, 1, 76, 15, 3, 6, 6, 4, 89, 2, 1

Offset: 1

Gus Wiseman, Mar 08 2025

nonn

First differs from A357982 at a(25) = 3, A357982(25) = 4.
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
A multiset partition can be regarded as an arrow in the ranked poset of integer partitions. For example, we have {{1},{1,2},{1,3},{1,2,3}}: {1,1,1,1,2,2,3,3} -> {1,3,4,6}, or (33221111) -> (6431) (depending on notation).
Set multipartitions are generally not transitive. For example, we have arrows: {{1},{1,2}}: {1,1,2} -> {1,3} and {{1,3}}: {1,3} -> {4}, but there is no set multipartition {1,1,2} -> {4}.

			The a(25) = 3 multisets are: {3,3}, {1,2,3}, {1,1,2,2}.

For constant instead of strict partitions see A381453, A355733, A381455, A000688.
Positions of 1 are A003586.
The upper version is A381078, before sums A050320.
For distinct block-sums see A381634, A381633, A381806.
Multiset partitions of prime indices:
- For multiset partitions (A001055) see A317141 (upper), A300383 (lower).
- For strict multiset partitions (A045778) see A381452.
- For set systems (A050326, zeros A293243) see A381441 (upper).
- For sets of constant multisets (A050361) see A381715.
- For strict multiset partitions with distinct sums (A321469) see A381637.
- For sets of constant multisets with distinct sums (A381635, zeros A381636) see A381716.
More on set systems: A050342, A116539, A296120, A318361.
More on set multipartitions: A089259, A116540, A270995, A296119, A318360.
More on set multipartitions with distinct sums: A279785, A381717, A381718.
A000041 counts integer partitions, strict A000009.
A000040 lists the primes.
A003963 gives product of prime indices.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A122111 represents conjugation in terms of Heinz numbers.
A265947 counts refinement-ordered pairs of integer partitions.
A358914 counts twice-partitions into distinct strict partitions.
Cf. A000720, A001222, A002846, A005117, A066328, A213242, A213385, A213427, A293511, A299200, A299201, A299202, A300385, A317142.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Union[Sort/@Join@@@Tuples[Select[IntegerPartitions[#],UnsameQ@@#&]&/@prix[n]]]],{n,100}]

a(A002110(n)) = A381808(n).

A381635 Number of ways to partition the prime indices of n into constant blocks with distinct sums.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 0, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 0, 1, 1, 0, 4, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 0, 1, 1, 1

Offset: 1

Gus Wiseman, Mar 09 2025

nonn

First differs from A381716 at a(1728) = 5, A381716(1728) = 4.

Also the number of factorizations on n into prime powers > 1 with distinct sums of prime indices (A056239).

			The a(432) = 3 multiset partitions:
  {{2,2,2},{1,1,1,1}}
  {{1},{1,1,1},{2,2,2}}
  {{1},{2},{2,2},{1,1,1}}
Note {{2},{2,2},{1,1,1,1}} is not included, as it does not have distinct block-sums.

Without distinct block-sums we have A000688, after sums A381455 (upper), A381453 (lower).
For distinct blocks instead of sums we have A050361, after sums A381715.
For strict instead of constant we have A381633 (zeros A381806), after sums A381634.
Positions of 0 are A381636.
Taking block-sums (and sorting) gives A381716.
Other multiset partitions of prime indices:
More on multiset partitions into constant blocks: A006171, A279784, A295935.
A001055 counts multiset partitions, see A317141 (upper), A300383 (lower).
A003963 gives product of prime indices.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A265947 counts refinement-ordered pairs of integer partitions.
Cf. A000720, A001222, A002846, A005117, A213242, A213385, A293511, A299202, A300385, A317142, A381870.

hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k]];
pfacs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#,d]&)/@Select[pfacs[n/d],Min@@#>=d&],{d,Select[Rest[Divisors[n]],PrimePowerQ]}]];
Table[Length[Select[pfacs[n],UnsameQ@@hwt/@#&]],{n,100}]

A381441 Number of multisets that can be obtained by partitioning the prime indices of n into a set of sets (set system) and taking their sums.

Original entry on oeis.org

1, 1, 1, 0, 1, 2, 1, 0, 0, 2, 1, 1, 1, 2, 2, 0, 1, 1, 1, 1, 2, 2, 1, 0, 0, 2, 0, 1, 1, 5, 1, 0, 2, 2, 2, 1, 1, 2, 2, 0, 1, 5, 1, 1, 1, 2, 1, 0, 0, 1, 2, 1, 1, 0, 2, 0, 2, 2, 1, 4, 1, 2, 1, 0, 2, 5, 1, 1, 2, 5, 1, 0, 1, 2, 1, 1, 2, 5, 1, 0, 0, 2, 1, 4, 2, 2, 2

Offset: 1

Gus Wiseman, Mar 06 2025

nonn

First differs from A050326 at a(210) = 13, A050326(210) = 15. This comes from the set systems {{3},{1,2,4}} and {{1,2},{3,4}}, and from {{4},{1,2,3}} and {{1,3},{2,4}}.
Also the number of multisets that can be obtained by taking the sums of prime indices of each factor in a strict factorization of n into squarefree numbers > 1.
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
A multiset partition can be regarded as an arrow in the ranked poset of integer partitions. For example, we have {{1},{1,2},{1,3},{1,2,3}}: {1,1,1,1,2,2,3,3} -> {1,3,4,6}, or (33221111) -> (6431) (depending on notation).
Sets of sets are generally not transitive. For example, we have arrows: {{1},{1,2}}: {1,1,2} -> {1,3} and {{1,3}}: {1,3} -> {4}, but there is no set of sets {1,1,2} -> {4}.

			The prime indices of 60 are {1,1,2,3}, with partitions into sets of sets:
  {{1},{1,2,3}}
  {{1,2},{1,3}}
  {{1},{2},{1,3}}
  {{1},{3},{1,2}}
with block-sums: {1,6}, {3,4}, {1,2,4}, {1,3,3}, which are all different, so a(60) = 4.

Before taking sums we had A050326, non-strict A050320.
Positions of 0 are A293243.
Positions of 1 are A293511.
This is the strict version of A381078 (lower A381454).
For distinct block-sums (instead of blocks) we have A381634, before sums A381633.
Other multiset partitions of prime indices:
- For multisets of constant multisets (A000688) see A381455 (upper), A381453 (lower).
- For multiset partitions (A001055) see A317141 (upper), A300383 (lower).
- For strict multiset partitions (A045778) see A381452.
- For sets of constant multisets (A050361) see A381715.
- For strict multiset partitions with distinct sums (A321469) see A381637.
- For sets of constant multisets with distinct sums (A381635) see A381716, A381636.
More on set systems: A050342, A116539, A279785, A296120, A318361.
A000041 counts integer partitions, strict A000009.
A000040 lists the primes, differences A001223.
A003963 gives product of prime indices.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A122111 represents conjugation in terms of Heinz numbers.
A265947 counts refinement-ordered pairs of integer partitions.
Cf. A000720, A001222, A002846, A005117, A066328, A213242, A213385, A213427, A299202, A300385, A317142.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Union[Sort[Total/@prix/@#]&/@Select[facs[n],UnsameQ@@#&&And@@SquareFreeQ/@#&]]],{n,100}]

a(A002110(n)) = A066723(n).

A381078 Number of multisets that can be obtained by partitioning the prime indices of n into a multiset of sets (set multipartition) and taking their sums.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 5, 1, 1, 2, 2, 2, 3, 1, 2, 2, 2, 1, 5, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 6, 1, 2, 2, 1, 2, 5, 1, 2, 2, 5, 1, 3, 1, 2, 2, 2, 2, 5, 1, 2, 1, 2, 1, 6, 2, 2, 2

Offset: 1

Gus Wiseman, Mar 05 2025

nonn

First differs from A050320 at a(210) = 13, A050320(210) = 15. This comes from the set multipartitions {{3},{1,2,4}} and {{1,2},{3,4}}, and from {{4},{1,2,3}} and {{1,3},{2,4}}.
Also the number of multisets that can be obtained by taking the sums of prime indices of each factor in a factorization of n into squarefree numbers > 1.
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
A multiset partition can be regarded as an arrow in the ranked poset of integer partitions. For example, we have {{1},{1,2},{1,3},{1,2,3}}: {1,1,1,1,2,2,3,3} -> {1,3,4,6}, or (33221111) -> (6431) (depending on notation).
Set multipartitions are generally not transitive. For example, we have arrows: {{1},{1,2}}: {1,1,2} -> {1,3} and {{1,3}}: {1,3} -> {4}, but there is no set multipartition {1,1,2} -> {4}.

			The prime indices of 60 are {1,1,2,3}, with set multipartitions:
  {{1},{1,2,3}}
  {{1,2},{1,3}}
  {{1},{1},{2,3}}
  {{1},{2},{1,3}}
  {{1},{3},{1,2}}
  {{1},{1},{2},{3}}
with block-sums: {1,6}, {3,4}, {1,1,5}, {1,2,4}, {1,3,3}, {1,1,2,3}, which are all different multisets, so a(60) = 6.

Robert Price, Table of n, a(n) for n = 1..1000

Before taking sums we had A050320, strict A050326 (zeros A293243), distinct sums A381633.
For distinct blocks we have A381441.
The lower version is A381454.
For distinct block-sums we have A381634.
Other multiset partitions of prime indices:
- For multisets of constant multisets (A000688) see A381455 (upper), A381453 (lower).
- For multiset partitions (A001055) see A317141 (upper), A300383 (lower).
- For strict multiset partitions (A045778) see A381452.
- For sets of constant multisets (A050361) see A381717.
- For strict multiset partitions with distinct sums (A321469) see A381637.
- For sets of constant multisets with distinct sums (A381635) see A381716, A381636.
More on set multipartitions: A089259, A116540, A270995, A296119, A318360.
A000041 counts integer partitions, strict A000009.
A000040 lists the primes.
A003963 gives product of prime indices.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A122111 represents conjugation in terms of Heinz numbers.
A265947 counts refinement-ordered pairs of integer partitions.
Cf. A000720, A001222, A002846, A005117, A025487, A066328, A213242, A213385, A213427, A299201, A299202, A300385.

hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k]];
sqfacs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#,d]&)/@Select[sqfacs[n/d],Min@@#>=d&],{d,Select[Rest[Divisors[n]],SquareFreeQ]}]];
Table[Length[Union[Sort[hwt/@#]&/@sqfacs[n]]],{n,100}]

a(A002110(n)) = A066723(n).

A381453 Number of multisets that can be obtained by choosing a constant integer partition of each prime index of n and taking the multiset union.

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 3, 1, 3, 2, 2, 2, 4, 3, 4, 1, 2, 3, 4, 2, 6, 2, 3, 2, 3, 4, 4, 3, 4, 4, 2, 1, 4, 2, 6, 3, 6, 4, 8, 2, 2, 6, 4, 2, 6, 3, 4, 2, 6, 3, 4, 4, 5, 4, 4, 3, 8, 4, 2, 4, 6, 2, 8, 1, 8, 4, 2, 2, 6, 6, 6, 3, 4, 6, 6, 4, 6, 8, 4, 2, 5, 2, 2, 6, 4, 4, 8

Offset: 1

Gus Wiseman, Mar 08 2025

nonn

First differs from A355733 and A355735 at a(21) = 6, A355733(21) = A355735(21) = 5.
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
A multiset partition can be regarded as an arrow in the ranked poset of integer partitions. For example, we have {{1},{1,2},{1,3},{1,2,3}}: {1,1,1,1,2,2,3,3} -> {1,3,4,6}, or (33221111) -> (6431) (depending on notation).
Multisets of constant multisets are generally not transitive. For example, we have arrows: {{1,1},{2}}: {1,1,2} -> {2,2} and {{2,2}}: {2,2} -> {4}, but there is no multiset of constant multisets {1,1,2} -> {4}.

			The a(21) = 6 multisets are: {2,4}, {1,1,4}, {2,2,2}, {1,1,2,2}, {2,1,1,1,1}, {1,1,1,1,1,1}.
The a(n) partitions for n = 1, 3, 7, 13, 53, 21 (G = 16):
  ()  (2)   (4)     (6)       (G)                 (42)
      (11)  (22)    (33)      (88)                (411)
            (1111)  (222)     (4444)              (222)
                    (111111)  (22222222)          (2211)
                              (1111111111111111)  (21111)
                                                  (111111)

Robert Price, Table of n, a(n) for n = 1..300

Positions of 1 are A000079.
The strict case is A008966.
Before sorting we had A355731.
Choosing divisors instead of constant multisets gives A355733.
The upper version is A381455, before taking sums A000688.
Multiset partitions of prime indices:
- For multiset partitions (A001055) see A317141 (upper), A300383 (lower).
- For strict multiset partitions (A045778) see A381452.
- For set multipartitions (A050320) see A381078 (upper), A381454 (lower).
- For set systems (A050326, zeros A293243) see A381441 (upper).
- For sets of constant multisets (A050361) see A381715.
- For strict multiset partitions with distinct sums (A321469) see A381637.
- For set systems with distinct sums (A381633, zeros A381806) see A381634.
- For sets of constant multisets with distinct sums (A381635, zeros A381636) see A381716.
More on multiset partitions into constant blocks: A006171, A279784, A295935.
A000041 counts integer partitions, strict A000009.
A000040 lists the primes.
A003963 gives product of prime indices.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A122111 represents conjugation in terms of Heinz numbers.
A265947 counts refinement-ordered pairs of integer partitions.
Cf. A000720, A000961, A001222, A002577, A018818, A213242, A213385, A213427, A275870, A299200, A300273, A300385.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Union[Sort/@Join@@@Tuples[Select[IntegerPartitions[#],SameQ@@#&]&/@prix[n]]]],{n,nn}]

a(A002110(n)) = A381807(n).

A213242 Number of generalizations of the partition 1^n.

Original entry on oeis.org

1, 2, 3, 6, 12, 31, 89, 303, 1119, 4649, 20572, 99241, 502622, 2725840, 15424019, 92211327, 571446565, 3716191974, 24920512847, 174169990243, 1251875604302, 9326245177768, 71241318920624, 562221733320241, 4535497053407716, 37677863148632647, 319551379756283637

Offset: 1

Alois P. Heinz, Jun 14 2012

nonn

Consider the ranked poset L(n) of partitions defined in A002846. Then a(n) is the total number of paths of all lengths 0,1,...,n-1 that start at any node in the poset and end at 1^n.

			For n=5 there are a(5) = 12 paths to 1^5 = 11111: 11111; 2111->11111; 221->2111->11111; 311->2111->11111; 32->221->2111->11111; 32->311->2111->11111; 41->221->2111->11111; 41->311->2111->11111; 5->32->221->2111->11111; 5->32->311->2111->11111; 5->41->221->2111->11111; 5->41->311->2111->11111.

Alois P. Heinz, Table of n, a(n) for n = 1..50
Olivier Gérard, The ranked posets L(2),...,L(8)

Cf. A002846, A213385, A213427, A213597.

b:= proc(l) option remember; local n, i, j, t; n:=nops(l);
      `if`(n<2, 1, `if`(l[n]=0, b(subsop(n=NULL, l)),
      add(`if`(l[i]=0, 0, add(b([seq(l[t]-`if`(t=1, l[t],
      `if`(t=i, 1, `if`(t=j and t=i-j, -2, `if`(t=j or t=i-j,
      -1, 0)))), t=1..n)]), j=1..i/2)), i=2..n)))
    end:
g:= proc(n, i, l)
      `if`(n=0 and i=0, b(l), `if`(i=1, b([n, l[]]), add(g(n-i*j, i-1,
      `if`(l=[] and j=0, l, [j, l[]])), j=0..n/i)))
    end:
a:= n-> g(n, n, []):
seq(a(n), n=1..25);

b[l_] := b[l] = With[{n = Length[l]}, If[n < 2, 1, If[l[[n]] == 0, b[ReplacePart[l, n -> Sequence[] ]], Sum[If[l[[i]] == 0, 0, Sum[b[Join[Table[l[[t]]-If[t == 1, l[[t]], If[t == i, 1, If[t == j && t == i-j, -2, If[t == j || t == i-j, -1, 0]]]], {t, 1, n}]]], {j, 1, i/2}]], {i, 2, n}]]] ]; g[n_, i_, l_] := If[n == 0 && i == 0, b[l], If[i == 1, b[Prepend[l, n]], Sum[g[n-i*j, i-1, If[l == {} && j == 0, l, Prepend[ l, j]]], {j, 0, n/i}]]] ; a[n_] := g[n, n, {}]; Table[a[n], {n, 1, 27}] // Flatten (* Jean-François Alcover, Dec 18 2013, translated from Maple *)

Edited by Alois P. Heinz at the suggestion of Gus Wiseman, May 02 2016

A381634 Number of multisets that can be obtained by taking the sum of each block of a set multipartition (multiset of sets) of the prime indices of n with distinct block-sums.

Original entry on oeis.org

1, 1, 1, 0, 1, 2, 1, 0, 0, 2, 1, 1, 1, 2, 2, 0, 1, 1, 1, 1, 2, 2, 1, 0, 0, 2, 0, 1, 1, 4, 1, 0, 2, 2, 2, 1, 1, 2, 2, 0, 1, 5, 1, 1, 1, 2, 1, 0, 0, 1, 2, 1, 1, 0, 2, 0, 2, 2, 1, 3, 1, 2, 1, 0, 2, 5, 1, 1, 2, 4, 1, 0, 1, 2, 1, 1, 2, 5, 1, 0, 0, 2, 1, 4, 2, 2, 2

Offset: 1

Gus Wiseman, Mar 06 2025

nonn

First differs from A050326 at a(30) = 4, A050326(30) = 5.
First differs from A339742 at a(42) = 5, A339742(42) = 4.
First differs from A381441 at a(30) = 4, A381441(30) = 5.
First differs from A381633 at a(210) = 10, A381633(210) = 12.
Also the number of multisets that can be obtained by taking the sums of prime indices of each factor in a factorization of n into squarefree numbers > 1 with distinct sums of prime indices (A056239).
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
A multiset partition con be regarded as an arrow in the ranked poset of integer partitions. For example, we have {{1},{1,2},{1,3},{1,2,3}}: {1,1,1,1,2,2,3,3} -> {1,3,4,6}, or (33221111) -> (6431) (depending on notation).
Set multipartitions with distinct block-sums are generally not transitive. For example, we have arrows: {{1},{1,2}}: {1,1,2} -> {1,3} and {{1,3}}: {1,3} -> {4}, but there is no arrow {1,1,2} -> {4}.

			The prime indices of 120 are {1,1,2,3}, with 3 ways:
  {{1},{1,2,3}}
  {{1,2},{1,3}}
  {{1},{2},{1,3}}
with block-sums: {1,6}, {3,4}, {1,2,4}, so a(120) = 3.
The prime indices of 210 are {1,2,3,4}, with 12 ways:
  {{1,2,3,4}}
  {{1},{2,3,4}}
  {{2},{1,3,4}}
  {{3},{1,2,4}}
  {{4},{1,2,3}}
  {{1,2},{3,4}}
  {{1,3},{2,4}}
  {{1},{2},{3,4}}
  {{1},{3},{2,4}}
  {{1},{4},{2,3}}
  {{2},{3},{1,4}}
  {{1},{2},{3},{4}}
with block-sums: {10}, {1,9}, {2,8}, {3,7}, {4,6}, {3,7}, {4,6}, {1,2,7}, {1,3,6}, {1,4,5}, {2,3,5}, {1,2,3,4}, of which 10 are distinct, so a(210) = 10.

Without distinct block-sums we have A381078 (lower A381454), before sums A050320.
For distinct blocks instead of sums we have A381441, before sums A050326, see A358914.
Before taking sums we had A381633.
Positions of 0 are A381806.
Positions of 1 are A381870, superset of A293511.
More on set multipartitions with distinct sums: A279785, A381717, A381718.
A001055 counts multiset partitions, see A317141 (upper), A300383 (lower).
A003963 gives product of prime indices.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A265947 counts refinement-ordered pairs of integer partitions.
Cf. A000720, A001222, A002846, A005117, A116540, A213242, A213385, A213427, A299202, A300385, A317142, A317143, A318360.

hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k]];
sfacs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#,d]&)/@Select[sfacs[n/d],Min@@#>=d&],{d,Select[Rest[Divisors[n]],SquareFreeQ]}]];
Table[Length[Union[Sort[hwt/@#]&/@Select[sfacs[n],UnsameQ@@hwt/@#&]]],{n,100}]

cp's OEIS Frontend

A002846 Number of ways of transforming a set of n indistinguishable objects into n singletons via a sequence of n-1 refinements.

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A265947 Total size of all principal order ideals in the poset of integer partitions of n with the refinement order.

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A213427 Number of ways of refining the partition n^1 to get 1^n.

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A381454 Number of multisets that can be obtained by choosing a strict integer partition of each prime index of n and taking the multiset union.

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A381635 Number of ways to partition the prime indices of n into constant blocks with distinct sums.

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A381441 Number of multisets that can be obtained by partitioning the prime indices of n into a set of sets (set system) and taking their sums.

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A381078 Number of multisets that can be obtained by partitioning the prime indices of n into a multiset of sets (set multipartition) and taking their sums.

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A381453 Number of multisets that can be obtained by choosing a constant integer partition of each prime index of n and taking the multiset union.

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